Toward the Classification of Scalar Nonpolynomial Evolution Equations:Polynomiality in Top Three Derivatives

TL;DR

This paper proves that certain integrable scalar evolution equations of order at least 7 are polynomial in their top three derivatives, revealing structural constraints linked to their integrability.

Contribution

It establishes polynomiality in the top derivatives for integrable equations and introduces a grading framework to analyze their scale homogeneity.

Findings

Integrable equations of order ≥7 are polynomial in the top three derivatives.

A grading in the polynomial algebra helps characterize integrable equations.

Integrable equations are scale homogeneous with respect to the introduced grading.

Abstract

We prove that arbitrary (nonpolynomial) scalar evolution equations of order $m\ge 7$, that are integrable in the sense of admitting the canonical conserved densities $\ro^{(1)}$, $\ro^{(2)}$, and $\ro^{(3)}$ introduced in [MSS,1991], are polynomial in the derivatives $u_{m-i}$ for $i=0,1,2.$ We also introduce a grading in the algebra of polynomials in $u_k$ with $k\ge m-2$ over the ring of functions in $x,t,u,...,u_{m-3}$ and show that integrable equations are scale homogeneous with respect to this grading.